# Quasi exactly solvable quantum mechanical systems

**Project level: **PhD, Masters

Quasi exact solvability is closely related to exact solvability. If all the eigenvalues of a quantum mechanical system are known together with the corresponding eigenfunctions the system is exactly solvable. In contrast a system is quasi exactly solvable if only a finite number (usually the lowest lying ones) of exact eigenvalues and eigenfunctions are known. Many quasi exactly solvable systems of physical significance have been identified either by algebraization or by deforming an exactly solvable systems with an addition of a higher order interaction term together with a compensation term. Although solutions to quasi exactly solvable systems of one degree of freedom are reasonably simple to obtain by i.e. the Bethe ansatz method among other methods it has been a challenge to find exact solutions (exact eigenvalues and eigenfunctions within the invariant subspace) to quasi exactly solvable systems of many degrees of freedom.

The overall objectives of this project are to investigate mathematical structures of the multi-particle quasi exactly solvable systems and develop techniques for solving these systems exactly. This is a project in collaboration with Professor Ryu Sasaki at Yukawa Institute for Theoretical Physics of Kyoto University. We are looking for students to join us to work on the project.